One interpretation of probabilistic thresholds, suggested by Krivelevich, Lee, and Sudakov [1]}, is as a measure of robustness. For example, \(K_n\)  is extremely robust with respect to containing a perfect matching since the threshold for this property in \(\mathbb {G}(n,p)\)  is \(\log n / n\) . Below this density w.h.p. isolated vertices, which are very simple local obstructions, begin to appear.
